Finite Group Databasebeta

← #6475412

Subgroups — #6475412

Order 512.

1772 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2#36625C211yesmaximal, characteristic
3#33218C214yesmaximal, complemented
4#30295C214yesmaximal, complemented
5#29300C214yesmaximal, complemented
6#29299C214yesmaximal, complemented
7#27955C212yesmaximal, complemented
8#27954C212yesmaximal, complemented
9#25818C212yesmaximal, complemented
10#25817C212yesmaximal, complemented
11#21943C214yesmaximal
12C4×#407C211yesmaximal, characteristic
13C2×#533C2211yescharacteristic
14C2×#533C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#501C2214yescomplemented
17C2×#407C2212yes—
18C2×C4×M16C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2857C2212yes—
22#2857C2212yes—
23#2855—22no—
24#2855—22no—
25#2851C2212yes—
26#2851C2212yes—
27#2851C2212yes—
28#2851C2212yes—
29#2848—22no—
30#2848—22no—
31C4×C22⋊Q8—22no—
32C4×C22⋊Q8—22no—
33C4×C22⋊Q8—22no—
34C4×C22⋊Q8—22no—
35#2296C2212yes—
36#2296C2212yes—
37#2285C2211yescharacteristic
38#2285C2211yescharacteristic
39#2280C2212yes—
40#2280C2212yes—
41#2085C2218yescomplemented
42#2084C2214yescomplemented
43#2083C2214yescomplemented
44#2059C2218yescomplemented
45#2058C2214yescomplemented
46#2057C2214yescomplemented
47#1993C2214yescomplemented
48#1992C2214yescomplemented
49#1942C2218yescomplemented
50#1941C2218yescomplemented
51#1854C2218yes—
52#1854C2218yes—
53#1854C2218yes—
54#1854C2218yes—
55#1847C2214yescomplemented
56#1809C2218yes—
57#1809C2218yes—
58#1749C2212yes—
59#1749C2212yes—
60C4×C22⋊C8C2212yes—
61C4×C22⋊C8C2212yes—
62#1728C2212yes—
63C24×C4D412yes—
64C24×C4C2311yescharacteristic
65C22×M16C2314yes—
66#533—44no—
67#533—44no—
68C2×C4⋊Q8C2312yes—
69C2×C4⋊Q8C2312yes—
70C2×#125C2312yes—
71C2×#125C2312yes—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×#117C2312yes—
81C2×#117C2312yes—
82C2×#117C2312yes—
83C2×#117C2312yes—
84C22×C42—44no—
85C22×C42D412yes—
86C22×C42D412yes—
87C22×C42D412yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#501—48no—
95C2×#107C2318yes—
96C2×#107C2318yes—
97C2×#106C2318yes—
98C2×#106C2318yes—
99#407Q8116yescomplemented
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C4×M16—44no—
103C2×#97C2314yes—
104C2×C4×C8C2314yes—
105#391—22no—
106#391—22no—
107#391—22no—
108#391—22no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389C2314yes—
114#389C2314yes—
115#389C2314yes—
116#389C2314yes—
117#387—24no—
118#387—24no—
119#387—24no—
120#387—24no—
121#385—24no—
122#385—24no—
123#385—24no—
124#385—24no—
125#385—24no—
126#385—24no—
127#384C2312yes—
128#384C2312yes—
129#382C2312yes—
130#382C2312yes—
131#381—22no—
132#381—22no—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380—22no—
139#380C2314yes—
140#380C2314yes—
141#380C2314yes—
142#380C2314yes—
143#379—22no—
144#379—22no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—22no—
152C4×C22⋊C4—22no—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#340C2318yes—
157#336C2314yes—
158#336C2314yes—
159C25C2×D411yescharacteristic
160C22×Q8—22no—
161C22×Q8—22no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×Q818yes—
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C2×M16—44no—
181C2×M16—44no—
182C2×M16C2×Q8116yes—
183C22×C8C2418yes—
184C4⋊Q8—28no—
185C4⋊Q8—44no—
186#125—28no—
187#125—44no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C22⋊Q8—48no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193C22⋊Q8—48no—
194C4×Q8—48no—
195C4×Q8—48no—
196#117—42no—
197#117—44no—
198#117—42no—
199#117—42no—
200#117—44no—
201#117—42no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4C2414yes—
215C2×C4⋊C4C2414yes—
216C2×C4⋊C4C2414yes—
217C2×C4⋊C4C2414yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—44no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2411yescharacteristic
246C2×C42C2411yescharacteristic
247C2×C42C2411yescharacteristic
248C2×C42C2411yescharacteristic
249#107—48no—
250#107—48no—
251#106—48no—
252#106—48no—
253C22⋊C8C2×Q8116yes—
254C22⋊C8C2×Q8116yes—
255#97—28no—
256C4×C8—44no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267C24—41no—
268C24—41no—
269C24—42no—
270C24—42no—
271C24C4⋊Q814yes—
272C24C22×D411yescharacteristic
273C24C22×D411yescharacteristic
274C24C22×D411yescharacteristic
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—44no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—416no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—432no—
317C22×C4C4⋊Q814yes—
318C22×C4C4⋊Q814yes—
319C22×C4C4⋊Q814yes—
320C22×C4C22×D412yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C22×D412yes—
326C22×C4C22×D412yes—
327C22×C4C22×Q814yes—
328C22×C4C22×Q814yes—
329C22×C4C22×Q814yes—
330C22×C4C2511yescharacteristic
331M16—416no—
332C2×C8—28no—
333C2×C8—28no—
334C2×C8C22×Q8116yes—
335C4⋊C4—24no—
336C4⋊C4—28no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—28no—
340C4⋊C4—24no—
341C4⋊C4—48no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—48no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—48no—
358C22⋊C4—48no—
359C22⋊C4—48no—
360C22⋊C4—48no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—44no—
371C42—44no—
372C42—44no—
373C42—44no—
374C42H(5,𝔽2)14yes—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—48no—
391C23—48no—
392C23C2×C4⋊Q812yes—
393C23C2×C4⋊Q812yes—
394C23C2×C4⋊Q812yes—
395C23C23×D411yescharacteristic
396Q8—88no—
397Q8—88no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4C2×C4⋊Q812yes—
438C2×C4C2×C4⋊Q812yes—
439C2×C4C2×C4⋊Q812yes—
440C2×C4#53414yes—
441C2×C4#57514yes—
442C2×C4#57514yes—
443C2×C4C23×Q812yes—
444C2×C4C2×H(5,𝔽2)12yes—
445C2×C4#58512yes—
446C8—216no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—416no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22C22×C4⋊Q811yescharacteristic
464C22C2×#53411yescharacteristic
465C22#344911yescharacteristic
466C22C2×#57512yes—
467C22C2×#57512yes—
468C4—24no—
469C4—24no—
470C4—24no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—42no—
475C4—84no—
476C4—84no—
477C4—84no—
478C4—84no—
479C4#350114yes—
480C2—42no—
481C2—42no—
482C2—42no—
483C2#3501111yescharacteristic
484C2#6066311yescharacteristic
485C2#6066311yescharacteristic
486C2#6075212yes—
487C2#6095312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #33218C2semidirect
4. #30295C2semidirect
5. #29300C2semidirect
6. #29299C2semidirect
7. #27955C2semidirect
8. #27954C2semidirect
9. #25818C2semidirect
10. #25817C2semidirect
16. C2×#501C22semidirect
41. #2085C22semidirect
42. #2084C22semidirect
43. #2083C22semidirect
44. #2059C22semidirect
45. #2058C22semidirect
46. #2057C22semidirect
47. #1993C22semidirect
48. #1992C22semidirect
49. #1942C22semidirect
50. #1941C22semidirect
55. #1847C22semidirect
99. #407Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#366252
#332182
#302952
#293002
#292992
#279552
#279542
#258182
#258172
#219432
C4×#4072